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Monday, June 30, 2008

Conmathematical Resolution of Russell's Paradox

Author: Kedar Joshi (1979- )

Russell's Paradox -

'A paradox uncovered by Bertrand Russell in 1901 that forced a reformulation of set theory. One version of Russell's paradox, known as the barber paradox, considers a town with a male barber who, every day, shaves every man who doesn't shave himself, and no one else. Does the barber shave himself ? The scenario as described requires that the barber shave himself if and only if he does not ! Russell's paradox, in its original form considers the set of all sets that aren't members of themselves. Most sets, it would seem, aren't members of themselves - for example, the set of elephants is not an elephant - and so could be said to be ""run-of-the-mill"". However, some ""self-swallowing"" sets do contain themselves as members, such as the set of all sets, or the set of all things except Julius Caesar, and so on. Clearly, every set is either run-of-the-mill or self-swallowing, and no set can be both. But then, asked Russell, what about the set S of all sets that aren't members of themselves ? Somehow, S is neither a member of itself nor not a member of itself.'

( See David Darling : The Universal Book of Mathematics, 2004 )

Conmathematical Resolution -

The term 'Conmathematics' means conceptual mathematics ( invented by Dr. Kedar Joshi ( b. 1979 ), Cambridge, UK ). It is a meta - mathematical system that defines the structure of superultramodern mathematics. It essentially involves a heavy or profound conceptual approach which is in striking contrast with the traditional symbolic or set theoretic approach.

Now conmathematically Russell's paradox is quite easy to resolve. The conmathematical resolution could be stated in just one sentence : As there is no barber who shaves every man who doesn't shave himself, and no one else, likewise there is no set of all sets that aren't members of themselves.

This sentence is justified or explained below.

Suppose there is a barber who shaves every man who doesn't shave himself, and no one else. Now the barber himself is a man and the supposition requires that the barber shave himself if and only if he does not ! This contradiction straightaway implies that the supposition is false. That is, there is no barber who shaves every man who doesn't shave himself, and no one else.

The justification of the sentence 'there is no set of all sets that aren't members of themselves' goes on similar lines. Conmathematial foundations of mathematics, being very profound and deep, easily absorb shocks of such fuzzy paradoxes, where the set theoretical foundations need to be reformulated.